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The circular lens of a telescope, a microscope, and even your eye acts as a hole, called an aperture, through which light is allowed to pass An aperture diffracts the light, just as a single slit does Alternating bright and dark rings occur with a circular aperture, as shown in Figure 19-15 The equation for an aperture is similar to that for a single slit However, an aperture has a circular edge rather than the two edges of a slit, so slit width, w, is replaced by aperture diameter, D, and an extra geometric factor of 122 enters the equation, resulting in x1 122 L/D When light from a distant star is viewed through the aperture of a telescope, the image is spread out due to diffraction If two stars are close enough together, the images may be blurred together, as shown in Figure 19-16 In 1879, Lord Rayleigh, a British physicist, mathematician, and Nobel prize winner, established a criterion for determining whether there is one or two stars in such an image The Rayleigh criterion states that if the center of the bright spot of one star s image falls on the first dark ring of the second, the two images are at the limit of resolution That is, a viewer will be able to tell that there are two stars rather than only one If two images are at the limit of resolution, how far apart are the objects Using the Rayleigh criterion, the centers of the bright spots of the two images are a distance of x1 apart Figure 19-16 shows that similar triangles can be used to find that xobj/Lobj x1/L Combining this with x1 122 L/D to eliminate x1/L, and solving for the separation distance between objects, xobj, the following equation can be derived Rayleigh Criterion xobj.

D The separation distance between objects that are at the limit of resolution is equal to 122, times the wavelength of light, times the distance from the circular aperture to the objects, divided by the diameter of the circular aperture

Figure 19-16 Similar-triangle geometry allows you to calculate the actual separation distance of objects The blue and red colors are used only for the purpose of illustration (Illustration not to scale)

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Diffraction in the eye In bright light the eye s pupil is about 3 mm in diameter The eye is most sensitive to yellow-green light where 550 nm So the Rayleigh criterion applied to the eye gives xobj 2 10 4 Lobj The distance between the pupil and retina is about 2 cm, so two barely resolved point sources would be separated by about 4 m on the retina The spacing between the light detectors, the cones, in the most sensitive part of the eye, the fovea, is about 2 m Thus, in the ideal case, the three adjacent cones would record light, dark, and light It would seem that the eye is ideally constructed If cones were closer together, they would see details of the diffraction pattern, not of the sources If cones were farther apart, they would not be able to resolve all possible detail Applying the Rayleigh criterion to find the ability of the eye to separate two distance sources shows that the eye could separate two automobile headlamps (15 m apart) at a distance of 7 km In practice, however, the eye is not limited by diffraction Imperfections in the lens and the liquid that fills the eye reduce the eye s resolution to about five times that set by the Rayleigh criterion Most people use their eyes for purposes other than resolving point sources For example, the eye seems to have a built-in ability to detect straight edges Many telescope manufacturers advertise that their instruments are diffraction limited That is, they claim that their telescopes can separate two point sources at the Rayleigh criterion To reach this limit they must grind the mirrors and lenses to an accuracy of one-tenth of a wavelength, or about 55 nm The larger the diameter of the mirror, the greater the resolution of the telescope Unfortunately, light from planets or stars must go through Earth s atmosphere The same variations in the atmosphere that cause stars to twinkle keep telescopes from reaching the diffraction limit Because the Hubble Space Telescope is above Earth s atmosphere, the resolution of its images is much better than those of larger telescopes on Earth s surface.

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